Subgroup ($H$) information
| Description: | $C_{14}$ |
| Order: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Index: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{rr}
159 & 0 \\
0 & 159
\end{array}\right), \left(\begin{array}{rr}
490 & 0 \\
0 & 490
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Frattini subgroup (hence characteristic and normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,7$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and central.
Ambient group ($G$) information
| Description: | $C_{245}\times \SL(2,3)$ |
| Order: | \(5880\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7^{2} \) |
| Exponent: | \(2940\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable.
Quotient group ($Q$) structure
| Description: | $A_4\times C_{35}$ |
| Order: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Exponent: | \(210\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \) |
| Automorphism Group: | $C_2\times C_{12}\times S_4$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Outer Automorphisms: | $C_2^2\times C_{12}$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_{42}\times A_4).C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{245}\times \SL(2,3)$ | |||
| Normalizer: | $C_{245}\times \SL(2,3)$ | |||
| Minimal over-subgroups: | $C_{98}$ | $C_{70}$ | $C_{42}$ | $C_{28}$ |
| Maximal under-subgroups: | $C_7$ | $C_2$ |
Other information
| Möbius function | not computed |
| Projective image | $A_4\times C_{35}$ |